Stochastic seismic inversion and Bayesian facies classification applied to porosity modeling and igneous rock identification

We apply stochastic seismic inversion and Bayesian facies classification for porosity modeling and igneous rock identification in the presalt interval of the Santos Basin. This integration of seismic and well-derived information enhances reservoir characterization. Stochastic inversion and Bayesian classification are powerful tools because they permit addressing the uncertainties in the model. We used the ES-MDA algorithm to achieve the realizations equivalent to the percentiles P10, P50, and P90 of acoustic impedance, a novel method for acoustic inversion in presalt. The facies were divided into five: reservoir 1,reservoir 2, tight carbonates, clayey rocks, and igneous rocks. To deal with the overlaps in acoustic impedance values of facies, we included geological information using a priori probability, indicating that structural highs are reservoir-dominated. To illustrate our approach, we conducted porosity modeling using facies-related rock-physics models for rock-physics inversion in an area with a well drilled in a coquina bank and evaluated the thickness and extension of an igneous intrusion near the carbonate-salt interface. The modeled porosity and the classified seismic facies are in good agreement with the ones observed in the wells. Notably, the coquinas bank presents an improvement in the porosity towards the top. The a priori probability model was crucial for limiting the clayey rocks to the structural lows. In Well B, the hit rate of the igneous rock in the three scenarios is higher than 60%, showing an excellent thickness-prediction capability.

Nonlinear joint PP-PS AVO inversion based on improved Bayesian inference and LSSVM

Multiwave seismic technology promotes the application of joint PP–PS amplitude versus offset (AVO) inversion;however conventional joint PP–PS AVO inversioan is linear based on approximations of the Zoeppritz equations for multiple iterations. Therefore the inversion results of P-wave, S-wave velocity and density exhibit low precision in the faroffset;thus, the joint PP–PS AVO inversion is nonlinear. Herein, we propose a nonlinear joint inversion method based on exact Zoeppritz equations that combines improved Bayesian inference and a least squares support vector machine (LSSVM) to solve the nonlinear inversion problem. The initial parameters of Bayesian inference are optimized via particle swarm optimization (PSO). In improved Bayesian inference, the optimal parameter of the LSSVM is obtained by maximizing the posterior probability of the hyperparameters, thus improving the learning and generalization abilities of LSSVM. Then, an optimal nonlinear LSSVM model that defi nes the relationship between seismic refl ection amplitude and elastic parameters is established to improve the precision of the joint PP–PS AVO inversion. Further, the nonlinear problem of joint inversion can be solved through a single training of the nonlinear inversion model. The results of the synthetic data suggest that the precision of the estimated parameters is higher than that obtained via Bayesian linear inversion with PP-wave data and via approximations of the Zoeppritz equations. In addition, results using synthetic data with added noise show that the proposed method has superior anti-noising properties. Real-world application shows the feasibility and superiority of the proposed method, as compared with Bayesian linear inversion.

Bayesian seismic multi-scale inversion in complex Laplace mixed domains

Seismic inversion performed in the time or frequency domain cannot always recover the long-wavelength background of subsurface parameters due to the lack of low-frequency seismic records. Since the low-frequency response becomes much richer in the Laplace mixed domains, one novel Bayesian impedance inversion approach in the complex Laplace mixed domains is established in this study to solve the model dependency problem. The derivation of a Laplace mixed-domain formula of the Robinson convolution is the first step in our work. With this formula, the Laplace seismic spectrum, the wavelet spectrum and time-domain reflectivity are joined together. Next, to improve inversion stability, the object inversion function accompanied by the initial constraint of the linear increment model is launched under a Bayesian framework. The likelihood function and prior probability distribution can be combined together by Bayesian formula to calculate the posterior probability distribution of subsurface parameters. By achieving the optimal solution corresponding to maximum posterior probability distribution, the low-frequency background of subsurface parameters can be obtained successfully. Then, with the regularization constraint of estimated low frequency in the Laplace mixed domains, multi-scale Bayesian inversion inthe pure frequency domain is exploited to obtain the absolute model parameters. The effectiveness, anti-noise capability and lateral continuity of Laplace mixed-domain inversion are illustrated by synthetic tests. Furthermore,one field case in the east of China is discussed carefully with different input frequency components and different inversion algorithms. This provides adequate proof to illustrate the reliability improvement in low-frequency estimation and resolution enhancement of subsurface parameters, in comparison with conventional Bayesian inversion in the frequency domain.

Bayesian Markov chain Monte Carlo inversion for anisotropy of PP-and PS-wave in weakly anisotropic and heterogeneous media

A single set of vertically aligned cracks embedded in a purely isotropic background may be con- sidered as a long-wavelength effective transversely iso- tropy （HTI） medium with a horizontal symmetry axis. The crack-induced HTI anisotropy can be characterized by the weakly anisotropic parameters introduced by Thomsen. The seismic scattering theory can be utilized for the inversion for the anisotropic parameters in weakly aniso- tropic and heterogeneous HTI media. Based on the seismic scattering theory, we first derived the linearized PP- and PS-wave reflection coefficients in terms of P- and S-wave impedances, density as well as three anisotropic parameters in HTI media. Then, we proposed a novel Bayesian Mar- kov chain Monte Carlo inversion method of PP- and PS- wave for six elastic and anisotropic parameters directly. Tests on synthetic azimuthal seismic data contaminated by random errors demonstrated that this method appears more accurate, anti-noise and stable owing to the usage of the constrained PS-wave compared with the standards inver- sion scheme taking only the PP-wave into account.

An improved four-dimensional variation source term inversion model with observation error regularization

Aiming at the Four-Dimensional Variation source term inversion algorithm proposed earlier,the observation error regularization factor is introduced to improve the prediction accuracy of the diffusion model,and an improved Four-Dimensional Variation source term inversion algorithm with observation error regularization(OER-4DVAR STI model)is formed.Firstly,by constructing the inversion process and basic model of OER-4DVAR STI model,its basic principle and logical structure are studied.Secondly,the observation error regularization factor estimation method based on Bayesian optimization is proposed,and the error factor is separated and optimized by two parameters:error statistical time and deviation degree.Finally,the scientific,feasible and advanced nature of the OER-4DVAR STI model are verified by numerical simulation and tracer test data.The experimental results show that OER-4DVAR STI model can better reverse calculate the hazard source term information under the conditions of high atmospheric stability and flat underlying surface.Compared with the previous inversion algorithm,the source intensity estimation accuracy of OER-4DVAR STI model is improved by about 46.97%,and the source location estimation accuracy is improved by about 26.72%.

Bayesian Rayleigh wave inversion with an unknown number of layers

Surface wave methods have received much attention due to their efficient, flexible and convenient characteristics. However, there are still critical issues regarding a key step in surface wave inversion. In most existing methods, the number of layers is assumed to be known prior to the process of inversion. However, improper assignment of this parameter leads to erroneous inversion results. A Bayesian nonparametric method for Rayleigh wave inversion is proposed herein to address this problem. In this method, each model class represents a particular number of layers with unknown S-wave velocity and thickness of each layer. As a result, determination of the number of layers is equivalent to selection of the most applicable model class. Regarding each model class, the optimization search of S-wave velocity and thickness of each layer is implemented by using a genetic algorithm. Then, each model class is assessed in view of its efficiency under the Bayesian framework and the most efficient class is selected. Simulated and actual examples verify that the proposed Bayesian nonparametric approach is reliable and efficient for Rayleigh wave inversion, especially for its capability to determine the number of layers.

Fransdimensional Bayesian inversion of timedomain airborne EM data

为改善反演效果,获得全局最小解,减小反演结果对初始模型的依赖程度,本文将变维数贝叶斯反演应用于时间域航空电磁数据反演。变维数贝叶斯反演方法在贝叶斯方法基础上利用可逆跳跃马尔科夫链蒙特卡洛方法(RJMCMC)实现反演模型层数的变化。这种方法根据建议分布并利用蒙特卡洛方法充分搜索模型空间进行随机采样。只统计同时满足数据拟合要求和接受概率的候选模型,受初始模型影响小,收敛稳定,反演结果可靠,最终可获得反演模型的概率分布和不确定度信息。由于实际飞行中发射源高度很难精确测量,因此本文在反演过程中将发射源高度分为不变和发射源高度变化两种情况。同时本文在电阻率先验概率密度函数中引入加权系数以调整对反演模型的约束强度,可有效地解决电阻率断面中间层反演效果不理想的问题。本文通过反演中心回线装置的H型和分离装置K型、HK型断面添加高斯噪声后的仿真数据以及实测数据,验证了变维数贝叶斯方法反演时间域航空电磁数据的有效性。

A Bayesian Inference Approach to Reduce Uncertainty in Magnetotelluric Inversion: A Synthetic Case Study

The deterministic geophysical inversion methods are dominant when inverting magnetotelluric data whereby its results largely depends on the assumed initial model and only a single representative solution is obtained. A common problem to this approach is that all inversion techniques suffer from non-uniqueness since all model solutions are subjected to errors, under-determination and uncertainty. A statistical approach in nature is a possible solution to this problem as it can provide extensive information about unknown parameters. In this paper, we developed a 1D Bayesian inversion code based Metropolis-Hastings algorithm whereby the uncertainty of the earth model parameters were quantified by examining the posterior model distribution. As a test, we applied the inversion algorithm to synthetic model data obtained from available literature based on a three layer model (K, H, A and Q). The frequency for the magnetotelluric impedance data was generated from 0.01 to 100 Hz. A 5% Gaussian noise was added at each frequency in order to simulate errors to the synthetic results. The developed algorithm has been successfully applied to all types of models and results obtained have demonstrated a good compatibility with the initial synthetic model data.

Geoacoustic Inversion for Bottom Parameters via Bayesian Theory in Deep Ocean

We develop a new approach to estimating bottom parameters based on the Bayesian theory in deep ocean. The solution in a Bayesian inversion is characterized by its posterior probability density （PPD）, which combines prior information about the model with information from an observed data set. Bottom parameters are sensitive to the transmission loss （TL） data in shadow zones of deep ocean. In this study, TLs of different frequencies from the South China Sea in the summer of 2014 are used as the observed data sets. The interpretation of the multidimensional PPD requires the calculation of its moments, such as the mean, covariance, and marginal distributions, which provide parameter estimates and uncertainties. Considering that the sensitivities of shallow- zone TLs vary for different frequencies of the bottom parameters in the deep ocean, this research obtains bottom parameters at varying frequencies. Then, the inversion results are compared with the sampling data and the correlations between bottom parameters are determined. Furthermore, we show the inversion results for multi- frequency combined inversion. The inversion results are verified by the experimental TLs and the numerical results, which are calculated using the inverted bottom parameters for different source depths and receiver depths at the corresponding frequency.

Zoeppritz-based AVO inversion using an improved Markov chain Monte Carlo method

The conventional Markov chain Monte Carlo （MCMC） method is limited to the selected shape and size of proposal distribution and is not easy to start when the initial proposal distribution is far away from the target distribution. To overcome these drawbacks of the conventional MCMC method, two useful improvements in MCMC method, adaptive Metropolis （AM） algorithm and delayed rejection （DR） algorithm, are attempted to be combined. The AM algorithm aims at adapting the proposal distribution by using the generated estimators, and the DR algorithm aims at enhancing the efficiency of the improved MCMC method. Based on the improved MCMC method, a Bayesian amplitude versus offset （AVO） inversion method on the basis of the exact Zoeppritz equation has been developed, with which the P- and S-wave velocities and the density can be obtained directly, and the uncertainty of AVO inversion results has been estimated as well. The study based on the logging data and the seismic data demonstrates the feasibility and robustness of the method and shows that all three parameters are well retrieved. So the exact Zoeppritz-based nonlinear inversion method by using the improved MCMC is not only suitable for reservoirs with strong-contrast interfaces and longoffset ranges but also it is more stable, accurate and antinoise.

Seismic AVO statistical inversion incorporating poroelasticity

Seismic amplitude variation with offset(AVO) inversion is an important approach for quantitative prediction of rock elasticity,lithology and fluid properties.With Biot-Gassmann's poroelasticity,an improved statistical AVO inversion approach is proposed.To distinguish the influence of rock porosity and pore fluid modulus on AVO reflection coefficients,the AVO equation of reflection coefficients parameterized by porosity,rock-matrix moduli,density and fluid modulus is initially derived from Gassmann equation and critical porosity model.From the analysis of the influences of model parameters on the proposed AVO equation,rock porosity has the greatest influences,followed by rock-matrix moduli and density,and fluid modulus has the least influences among these model parameters.Furthermore,a statistical AVO stepwise inversion method is implemented to the simultaneous estimation of rock porosity,rock-matrix modulus,density and fluid modulus.Besides,the Laplace probability model and differential evolution,Markov chain Monte Carlo algorithm is utilized for the stochastic simulation within Bayesian framework.Models and field data examples demonstrate that the simultaneous optimizations of multiple Markov chains can achieve the efficient simulation of the posterior probability density distribution of model parameters,which is helpful for the uncertainty analysis of the inversion and sets a theoretical fundament for reservoir characterization and fluid discrimination.

All-parameters Rayleigh wave inversion

Since S-wave velocity of the subsurface is an important parameter in near surface applications,many studies have been conducted for its estimation.Among the various methods that use surface waves or body waves,Rayleigh wave inversion is the most popular.In practice,the densities and P-wave velocities of different layers are usually assumed to be known to avoid ill-posed problems,as they have less influence on the dispersion curves.However,improper assignment of these two groups of parameters leads to inaccurate estimation of the S-wave velocity profile.In order to address this problem,the all-parameters Rayleigh wave inversion strategy is proposed in which the S-wave velocities,layer thicknesses,densities and P-wave velocities of different layers are included as the unknown parameters for inversion.Meanwhile,the transitional Markov Chain Monte Carlo(TMCMC)algorithm is applied for the implementation of all-parameters Rayleigh wave inversion.One simulated example and two real-test applications are demonstrated to verify the capability of the proposed method in the estimation of the S-wave velocity profile,the densities and the P-wave velocities.Furthermore,it is verified that the proposed method achieved more accurate S-wave velocity profile estimation than the traditional approach.

Entropy Bayesian Analysis for the Generalized Inverse Exponential Distribution Based on URRSS

This paper deals with the Bayesian estimation of Shannon entropy for the generalized inverse exponential distribution.Assuming that the observed samples are taken from the upper record ranked set sampling(URRSS)and upper record values(URV)schemes.Formulas of Bayesian estimators are derived depending on a gamma prior distribution considering the squared error,linear exponential and precautionary loss functions,in addition,we obtain Bayesian credible intervals.The random-walk Metropolis-Hastings algorithm is handled to generate Markov chain Monte Carlo samples from the posterior distribution.Then,the behavior of the estimates is examined at various record values.The output of the study shows that the entropy Bayesian estimates under URRSS are more convenient than the other estimates under URV in the majority of the situations.Also,the entropy Bayesian estimates perform well as the number of records increases.The obtained results validate the usefulness and efficiency of the URV method.Real data is analyzed for more clarifying purposes which validate the theoretical results.

Comparison of deterministic and stochastic approaches to crosshole seismic travel-time inversions

The Bayesian inversion method is a stochastic approach based on the Bayesian theory.With the development of sampling algorithms and computer technologies,the Bayesian inversion method has been widely used in geophysical inversion problems.In this study,we conduct inversion experiments using crosshole seismic travel-time data to examine the characteristics and performance of the stochastic Bayesian inversion based on the Markov chain Monte Carlo sampling scheme and the traditional deterministic inversion with Tikhonov regularization.Velocity structures with two different spatial variations are considered,one with a chessboard pattern and the other with an interface mimicking the Mohorovicicdiscontinuity(Moho).Inversions are carried out with different scenarios of model discretization and source–receiver configurations.Results show that the Bayesian method yields more robust single-model estimations than the deterministic method,with smaller model errors.In addition,the Bayesian method provides the posterior probabilistic distribution function of the model space,which can help us evaluate the quality of the inversion result.

Complex spherical-wave elastic inversion using amplitude and phase reflection information

Unlike the real-valued plane wave reflection coefficient(PRC)at the pre-critical incident angles,the frequency-and depth-dependent spherical-wave reflection coefficient(SRC)is more accurate and always a complex value,which contains more reflection amplitude and phase information.In near field,the imaginary part of complex SRC(phase)cannot be ignored,but it is rarely considered in seismic inversion.To promote the practical application of spherical-wave seismic inversion,a novel spherical-wave inversion strategy is implemented.The complex-valued spherical-wave synthetic seismograms can be obtained by using a simple harmonic superposition model.It is assumed that geophone can only record the real part of complex-valued seismogram.The imaginary part can be further obtained by the Hilbert transform operator.We also propose the concept of complex spherical-wave elastic impedance(EI)and the complex spherical-wave EI equation.Finally,a novel complex spherical-wave EI inversion approach is proposed,which can fully use the reflection information of amplitude,phase,and frequency.With the inverted complex spherical-wave EI,the velocities and density can be further extracted.Synthetic data and field data examples show that the elastic parameters can be reasonably estimated,which illustrate the potential of our spherical-wave inversion approach in practical applications.

Inverse Bayesian Estimation of Gravitational Mass Density in Galaxies from Missing Kinematic Data

In this paper, we focus on a type of inverse problem in which the data are expressed as an unknown function of the sought and unknown model function (or its discretised representation as a model parameter vector). In particular, we deal with situations in which training data are not available. Then we cannot model the unknown functional relationship between data and the unknown model function (or parameter vector) with a Gaussian Process of appropriate dimensionality. A Bayesian method based on state space modelling is advanced instead. Within this framework, the likelihood is expressed in terms of the probability density function (pdf) of the state space variable and the sought model parameter vector is embedded within the domain of this pdf. As the measurable vector lives only inside an identified sub-volume of the system state space, the pdf of the state space variable is projected onto the space of the measurables, and it is in terms of the projected state space density that the likelihood is written;the final form of the likelihood is achieved after convolution with the distribution of measurement errors. Application motivated vague priors are invoked and the posterior probability density of the model parameter vectors, given the data are computed. Inference is performed by taking posterior samples with adaptive MCMC. The method is illustrated on synthetic as well as real galactic data.

基于DREAM_ZS算法的EIT电阻率反演方法研究

针对电阻抗成像(EIT)中的电阻率反演及其不确定性量化问题,提出基于贝叶斯理论的不确定性分析方法.首先,利用反向传播(BP)神经网络模型作为正问题替代模型,取得了计算精度高的结果,并且大大提高计算效率.然后,采用基于贝叶斯理论的自适应差分进化Metropolis抽样(DREAM_ZS)算法对电阻率进行反演,并对不同激励模式和不同先验分布进行了对比分析.对模拟头部的4层同心圆模型的反演结果显示,DREAM_ZS抽样算法能够对4个参数进行准确识别,相对激励模式的反演效果最优.4个参数的不确定性程度不同,头皮电阻率不确定性最小,敏感性最强,其次是颅骨,大脑和脑脊液的不确定性较大.进而,对高维参数的圆模型进行仿真,采用相对激励模式,DREAM_ZS抽样算法能够准确反演二维圆模型的各个参数.参数的先验分布为正态分布时,与均匀分布相比,其反演结果不确定性小,对算法的识别效果更强.

2022年门源M_(S)6.9地震的同震滑动分布:联合InSAR、GPS和地表位错的贝叶斯建模

2022年1月8日青海省门源M_(S)6.9地震发生于托莱山断裂东段和冷龙岭断裂西段的交汇部位,其运动学特征关系到青藏高原北缘的动力学行为,同时也是区域地震危险性评估中的基础数据.本研究针对该地震,综合InSAR和GPS观测的同震位移场,以及野外地质调查的同震位错数据,采用贝叶斯反演方法,构建同震滑动分布,断层模型采用三角网格更好地刻画断层的不规则几何形态.结果显示,本次左旋走滑型地震的主要滑动发生在冷龙岭断裂西段,深度范围约2~6 km,计算的地震矩为0.95×10^(19)N·m,对应矩震级M_(W)6.65.利用多数据的联合反演解析出,沿走向存在两个滑动集中区,东侧的最大滑动达约4.8 m.震间亏损能量和库仑应力分析表明,托莱山断裂和冷龙岭断裂仍存在未来发震潜能;该地震造成的应力变化,对长期地震空区金强河—毛毛山段落有加载作用,因此,该段未来地震危险性值得关注.

岩石物理驱动的储层物性参数非线性地震反演方法

叠前地震反演和岩石物理反演分别是获取弹性参数和物性参数的重要手段,两者结合有助于实现储层参数预测并精细刻画储层特征.储层物性参数的反演依赖于岩石物理模型,在进行物性参数反演时可以将复杂的岩石物理模型做泰勒展开,进而得到其一阶或高阶的近似表达式,然而这会降低模型的精确性并增加反演的误差.为了提高储层物性参数反演的稳定性和准确性,本文以碎屑岩储层为例,提出了岩石物理驱动的储层物性参数非线性地震反演方法.首先,基于贝叶斯框架和高斯分布约束条件,从叠前地震数据中实现纵、横波速度及密度等弹性参数的反演.其次,通过碎屑岩岩石物理模型建立起弹性参数与物性参数之间的联系.最后,利用粒子群算法进行全局寻优获得较为准确的孔隙度、泥质含量和含水饱和度等物性参数.合成数据和实际资料测试结果验证了所提方法的可行性和准确性,反演结果与测井数据吻合较好,可有效指示含气储层区域,本文方法在储层预测和评价方面具有广泛的应用前景.

基于精确Zoeppritz方程的贝叶斯叠前地震随机反演

基于精确Zoeppritz方程的叠前地震反演方法在面向低信噪比地震资料的应用时仍然存在较大挑战。马尔科夫链蒙特卡洛(Markov chain Monte Carlo, MCMC)模拟是一种启发式的全局寻优算法,是实现叠前弹性参数非线性反演的有效途径。常规基于MCMC算法的叠前反演采用高斯分布来刻画弹性参数的统计特征,在应用于复杂岩性储层时有较大的局限性。同时,由于地下模型参数空间巨大以及地震数据中噪声等因素的影响,MCMC对弹性参数后验概率分布的搜索过程极易受到局部极值的影响,这使得基于MCMC的叠前反演较难获得稳定、准确的结果。本文针对实际复杂储层及低信噪比地震资料条件下基于精确Zoeppritz方程的叠前反演问题,提出了一种改进的MCMC弹性参数反演方法。该方法首先利用低频模型约束,将待反演参数转换为模型参数的扰动量,从而降低后验概率分布的复杂度;其次,通过对似然函数取对数,并利用低频模型来约束地震正演过程;最后,利用基于随机子空间采样的多链算法对叠前非线性反演问题进行全局寻优,以避免采样过程过早地收敛到局部极值。低信噪比模拟数据和实际数据的测试表明,本文所提方法能够获得更加准确、稳定的弹性参数反演结果,并且能够对反演结果给出可信、定量的不确定性估计。